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# Lecture 2: Exercises

### Suggestions for further reading

- Convergence tests: Arfken (chaps. 5.2, 5.3)

### Questions for Review

- Which are the most important convergence tests for infinite series?
- What is the comparison test?
- What is the Cauchy root test?
- How is the Cauchy root test related to the comparison test?
- What is the D'Alambert ratio test? How is it related to the comparison
test?
- What is the Leibnitz criterion? To which kind of infinite series does
it apply? Give an example!
- How is Euler's number e defined? What is (approximately) its numerical
value?
- How can functions be represented?
- What does it mean to say that a function is
- bounded?
- positive or negative definite?
- even or odd?
- periodic?
- (strictly) monotonic?

- How is the inverse of a function defined?
- How can the inverse be constructed graphically?
- Do all functions have an inverse? Give examples!
- Do all strictly monotonic functions have an inverse?
- Are all invertible functions strictly monotonic?
- Are the functions x^2 and cos(x) invertible? If so, why? If not, what
can one do in order to make them invertible?
- When is a function continuous?
- Which are the most common types of discontinuities?
- What is the Heaviside step function? Is it continuous?
- Can functions have a well-defined limit at a gap? If so, how is this
formulated mathematically?
- What is the limit of sin(x)/x as x goes to zero?
- Give examples for (nontrivial) functions that approach an asymptotic
value as the argument goes to plus or minus infinity. How is this formulated
mathematically?

### Problems